= Solution
For each fixed finite tuple length $n$, use the finite <conjunction>
$$
\boxed{\operatorname{LI}_n(\bar x)=\bigwedge_{\bar a\in K^n\setminus\{0\}}\left(\sum_{j=1}^n\lambda_{a_j}(x_j)\ne0\right).}
$$
It asserts that no nontrivial <scalar multiplication>[scalar] combination vanishes, exactly <linear independence>. It is an $L$-formula because the <field> is finite and every <scalar multiplication> is named. <scalar multiplication>[Scalar coefficients] are indices of <function symbols>, not variables ranging over a separate <field> sort. There is no single finite formula with arbitrarily many <free variables>; the assertion is a formula for each prescribed finite $n$.
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